Finite Group DatabasebetaFinite Group Databasebeta| order | 512 |
| nilpotency class | 4 |
| isoclinism | isoclinic class of #500743 |
| automorphism | ? |
| homology | Abelianization (H1) = C24 · Schur multiplier (H2) = C26 |
No aliases for this group.
| subgroup | G/H | |
|---|---|---|
| center, socle | C2 | #24248 |
| commutator, Frattini, cosocle | C22×C8 | C24 |
| LCS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
| UCS | C1 ◁ C2 ◁ C22×C4 ◁ #2929 ◁ G | |
| LECS | G ▷ C22×C8 ▷ C4 ▷ C2 ▷ C1 | |
1028 conjugacy classes of subgroups.
| minimal degree | μ(G) = 64 |
| transitive degree | μt(G) = 64 |
A minimal faithful permutation representation — 4 generators on 64 points:
(1,2,8,14)(3,9,19,43)(4,20,23,51)(5,27,31,12)(6,32,30,11)(7,36,29,13)(10,34,46,24)(15,55,56,64)(16,41,33,52)(17,40,35,50)(18,53,37,42)(21,45,61,59)(22,44,58,60)(25,48,28,54)(26,47,38,49)(39,62,63,57);(1,3)(2,9)(4,21,23,61)(5,17)(6,33)(7,37)(8,19)(10,44,46,60)(11,41)(12,50)(13,53)(14,43)(15,25,56,28)(16,30)(18,29)(20,59,51,45)(22,24,58,34)(26,57,38,62)(27,40)(31,35)(32,52)(36,42)(39,47,63,49)(48,55,54,64);(1,4)(2,10)(3,15)(5,28)(6,34)(7,38)(8,23)(9,39)(11,49)(12,51)(13,54)(14,46)(16,21)(17,57)(18,22)(19,56)(20,27)(24,30)(25,31)(26,29)(32,47)(33,61)(35,62)(36,48)(37,58)(40,44)(41,55)(42,45)(43,63)(50,60)(52,64)(53,59);(1,3)(2,9)(4,22,23,58)(5,16)(6,35)(7,18)(8,19)(10,45,46,59)(11,40)(12,52)(13,42)(14,43)(15,38,56,26)(17,30)(20,60,51,44)(21,24,61,34)(25,62,28,57)(27,41)(29,37)(31,33)(32,50)(36,53)(39,54,63,48)(47,64,49,55)Polycyclic presentation:
2^9|a^2=i,b^2=e,d^2=e,f^2=hi,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=ghi,h^a=hi,c^b=cg,f^b=fh,g^b=gh,h^b=hi,d^c=dh,e^c=ei,f^c=fh,g^c=ghi,h^c=hi,g^d=gi50 conjugacy classes — view the full character table.
| maximal subgroup of | #72249754, #72249766, #72249836, … |
| maximal normal subgroup of | #72249754, #72249766, #72249836, … |
| maximal quotient of | #72249947, #72250091, #72250142, … |
Construct this group in GAP via the SmallGroups library:
gap> SmallGroup(512, 407941);